What is the period of $y = \tan \frac{x}{2}$?
Explanation: The graph of $y=\tan \frac{x}{2}$ passes through one full period as $\frac{x}{2}$ ranges from $-\frac{\pi}{2}$ to $\frac{\pi}{2},$ which means $x$ ranges from $-\pi$ to $\pi.$  Thus, the period is $\pi - (-\pi) = \boxed{2 \pi}.$

The graph of $y=\tan \frac{x}{2}$ is shown below:

[asy]import TrigMacros;

size(400);

real g(real x)
{
	return tan(x/2);
}

draw(graph(g,-3*pi + 0.01,-pi - 0.01),red);
draw(graph(g,-pi + 0.01,pi - 0.01),red);
draw(graph(g,pi + 0.01,3*pi - 0.01),red);
limits((-3*pi,-5),(3*pi,5),Crop);
draw((-pi,-5)--(-pi,5),dashed);
draw((pi,-5)--(pi,5),dashed);
trig_axes(-3*pi,3*pi,-5,5,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);
[/asy]